Twin-Wire Networks for Zero Interconnect, High-Density 4-Wire Electrical Characterizations of Materials

Four-wire measurements have been introduced by Lord Kelvin in 1861 and have since become the standard technique for characterizing small resistances and impedances. However, high-density 4-wire measurements are generally complex, time-consuming, and inefficient because of constraints on interconnects, pads, external wires, and mechanical contacts, thus reducing reproducibility, statistical significance, and throughput. Here, we introduce, systematically design, analyze, and experimentally validate zero interconnect networks interfaced to external instrumentation by couples of twin wire. 3D-printed holders with magnets, interconnects, nonadhesive layers, and spacers can effortlessly establish excellent electrical connections with tunable or minimum contact forces and enable accurate measurements even for delicate devices, such as thin metals on soft polymers. As an example, we measured all the resistances of a twin-wire 29-resistor network made of silver-nanoparticle ink printed on polyimide, paper, or photo paper, including during sintering or temperature calibration, resulting in an unprecedentedly easy and accurate characterization of both resistivity and its temperature coefficient. The theoretical framework and experimental strategies reported here represent a breakthrough toward zero interconnect, simple, and efficient high-density 4-wire characterizations, can be generalized to other 4-wire measurements (impedances, sensors) and can open the way to more statistically meaningful and reproducible analyses of materials, high-throughput measurements, and minimally invasive characterizations of biomaterials.


Note S1 -Theoretical analysis of iterative multiply-by-M expansion procedures
In case of bifurcation (M = 2), starting with a single initial resistor, the number of additional resistors generated by bifurcation is equal to 4 during the first step and, for the following steps, to twice the number of resistors generated in the previous step, thus resulting in   In the more general case of iterative multiply-by-M expansion procedures (M = 2, 3, 4, …) the number of resistors generated in a certain iteration step is equal to 2M during the first step and, for the following multiply-by-M steps, is equal to M times the number of resistors generated in the previous step, so that Since the sum of the first N terms of a geometric series is it is convenient to rewrite the number of resistors NR as 10 11 1 1 2 1 2 1 which is obviously positive (M > 1).
Similarly, the number of pads is equal to 2M after the first expansion step and, for the other cases, is equal to M times the number of pads generated in the previous step, so that 2 S P NM  and, therefore (twin-wires), The ability of a device to enable the accurate measurement of many resistors with a small number of pads and, therefore, with a small number of wires (twin-wires), can be quantified by the ratio As shown in Figure 1h, this ratio, when the iteration index is increased, quickly tends to

Note S2 -Consequences of open-circuits in twin-wire resistive networks
Thin metal Fig. S3f), all the 6 residual resistors can be individually measured.

Note S3 -Estimation of the pad internal parasitic resistance
The order of magnitude of the worst case pad internal resistance can be roughly estimated as where XL, t and XW are the length, thickness and width, respectively, of a hypothetical resistor ( Figure 2c) and m is the electrical resistivity of the thin metal film.
In case of deposited gold thin films with thicknesses and pad lateral dimensions in the order of nanometers and millimeters, respectively, it may be convenient to rewrite

Note S4 -Validation experiments (repeated placements-removals and 180° rotations)
The mean value, maximum resistance variations and relative resistance changes during 8 repeated placements-removals were around (35.6 Ω, ± 54 mΩ, ± 0.15 %), (19.6 Ω, ± 120 mΩ, ± 0.6 %) and (21.3 Ω, ± 150 mΩ, ± 0.7 %) for RA, RB + RC, and RD + RE, respectively. These small changes can be attributed to several non-idealities. The expected relative resistance changes of gold resistors for a ± 0.5 K temperature difference is in the order of ± 0.12 % (the temperature coefficient of thin gold metal resistances may be around 2500 ppm/K). Moreover, voltage errors introduced by the input offset voltage of the instrumentation amplifier or by spurious Seebeck voltages also introduce resistance errors (e.g. a ± 50 µV voltage error, with 1 mA reading current, translates into a ± 50 mΩ resistance error). However, clearly, the variations for RA are smaller than for both RB + RC and RD + RE. In fact, RA is, in practice, measured by using one distinct wire for each pad or, equivalently, with a conventional method (1-wire-perpad). By contrast, the measurements of both RB + RC and RD + RE require the twin-wire strategy and, therefore, are affected by the parasitic pad resistances RPAD. Though the initial values of RPAD are very small (e.g. in the order of 0.12 Ω for our devices, 100 nm) and are anyway associated to the metal film to be characterized, RPAD is different for different pads, and, even for a given pad, at different iterations, can change due to slight misplacements and, more importantly, to eventual damages which can even indefinitely increase (up to open circuit) RPAD.
As a consequence, these experiments, besides confirming that the anti-adhesion layers effectively protect the device (otherwise all the resistances would increase), also allow to conclude that the pads are not significantly damaged by the measurement system, even after many connections and disconnections, because significant damages in a certain pad would immediately translate into irreversible and large increases of the correspondent (RB + RC or RD + RE, depending on the damaged pad) measurements.

Note S5 -Measurements of 29R twin-wire network
The system for measuring the zero-interconnects twin-wire 29-resistors network (Figure 3a-c,   Fig. S9) sequentially measures all the 29 resistances and, in order to simplify the synchronization of the multimeter data acquisition with the control PCB, also includes, as a mark, an additional, easily recognizable, measurement (i.e. many resistors in series, so that the measured resistance is larger than all the other measurements).
The complete data supporting this study, including all the files needed for asking a PCB company to replicate the hardware, the software and the raw data required to reproduce these findings are available to download from https://data.mendeley.com/datasets/pywhr745ns/1 (DOI: 10.17632/pywhr745ns.1). Figure S1. Network of NR resistors with NR + 1 pads and NR + 3 wires.             or Figure S2b).